Three‐Phase Barker Arrays

A 3‐phase Barker array is a matrix of third roots of unity for which all out‐of‐phase aperiodic autocorrelations have magnitude 0 or 1. The only known truly two‐dimensional 3‐phase Barker arrays have size 2 × 2 or 3 × 3. We use a mixture of combinatorial arguments and algebraic number theory to establish severe restrictions on the size of a 3‐phase Barker array when at least one of its dimensions is divisible by 3. In particular, there exists a double‐exponentially growing arithmetic function T such that no 3‐phase Barker array of size s × t with 3 ∣ t exists for all t < T ( s ) . For example, T ( 5 ) = 4860 , T ( 10 ) > 10 11 , and T ( 20 ) > 10 214 . When both dimensions are divisible by 3, the existence problem is settled completely: if a 3‐phase Barker array of size 3 r × 3 q exists, then r = q = 1 .